Large Deviations of Surface Height in the Kardar-Parisi-Zhang Equation

Baruch Meerson, Eytan Katzav, Arkady Vilenkin

Published 2015-12-15Version 1

We evaluate the probability distribution $\mathcal{P}(H)$ of large deviations $H$ of the evolving surface height $h(x,t)$ in the Kardar-Parisi-Zhang (KPZ) equation in one dimension when starting from a flat interface. We also determine the optimal history of the interface, conditioned on reaching the height $H$ at time $t$. The tails of $\mathcal{P}(H)$ behave, at arbitrary time $t>0$, and in a proper moving frame, as $-\ln \mathcal{P}(H)\sim |H|^{5/2}$ and $\sim |H|^{3/2}$. The $3/2$ tail coincides with the asymptotic of the Gaussian orthogonal ensemble Tracy-Widom distribution, previously only observed at long times.